Stationary measures for SL₂(ℝ)-actions on homogeneous bundles over flag varieties

Let 𝐺 be a real semisimple Lie group with finite centre and without compact factors, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Q</m:mi> <m:mo><</m:mo> <m:mi>G</m:mi> </m:mrow> </m:math> Q<G a parabolic subgroup and 𝑋 a homogeneous space of 𝐺 admitting an equivariant projection on the flag variety <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:mi>Q</m:mi> </m:mrow> </m:math> G/Q with fibres given by copies of lattice quotients of a semisimple factor of 𝑄. Given a probability measure 𝜇, Zariski-dense in a copy of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>H</m:mi> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mi>SL</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="double-struck">R</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> H=SL₂(R) in 𝐺, we give a description of 𝜇-stationary probability measures on 𝑋 and prove corresponding equidistribution results. Contrary to the results of Benoist–Quint corresponding to the case <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi> <m:mo>=</m:mo> <m:mi>Q</m:mi> </m:mrow> </m:math> G=Q , the type of stationary measures that 𝜇 admits depends strongly on the position of 𝐻 relative to 𝑄. We describe possible cases and treat all but one of them, among others using ideas from the works of Eskin–Mirzakhani and Eskin–Lindenstrauss.